I've been scratching my head for a while in order to somehow manage to make an improvised weapon magical (to overcome resistance/immunity to nonmagical attacks). I believe I've found a way, but I want to verify it here first.
First, here are the relevant Way of the Kensei monk features (Xanathar's Guide to Everything, p. 34-35):
[3rd level:] Kensei Weapons. Choose two types of weapons to be your kensei weapons: one melee weapon and one ranged weapon. Each of these weapons can be any simple or martial weapon that lacks the heavy and special properties. The longbow is also a valid choice.
[…]
[6th level:] Magic Kensei Weapons. Your attacks with your kensei weapons count as magical for the purpose of overcoming resistance and immunity to nonmagical attacks and damage.
Now, here are the relevant Improvised Weapons rules:
Sometimes characters don't have their weapons and have to attack with whatever is at hand. An improvised weapon includes any object you can wield in one or two hands, such as broken glass, a table leg, a frying pan, a wagon wheel, or a dead goblin.
Often, an improvised weapon is similar to an actual weapon and can be treated as such. For example, a table leg is akin to a club. At the DM's option, a character proficient with a weapon can use a similar object as if it were that weapon and use his or her proficiency bonus.
Now, let's say you have a level 6 Kensei monk that chose "club" as one of its Kensei weapons, and wields a table leg. When the monk hits an enemy with magic resistance (or immunity) to nonmagical attacks, can his damage be considered magical (if the DM agrees that the table leg works as a club)?
Note: I am fully aware that is it not optimal, but the build I have in mind involves 6 levels of Kensei monk and at least 6 levels of Tundra Storm herald barbarian in order to freeze water into ice, then use the ice as improvised weaponry (but treat the improvised weapon as a proper weapon chosen as kensei weapon in order to overcome nonmagical damage resistance/immunity). Now, determining which weapon is closest to a block of ice (in order to select it as the kensei weapon) is another set of gloves.
Best Answer
In your scenario, yes, a table leg can be treated as a Kensai weapon and thus overcome magical resistance.
In another situation, possibly not:
An improvised weapon must be similar enough to a weapon on the simple or martial table (DM ruling) or else it would not work as you say.
As to your note added in the edit: Yes. A block of ice frozen in the shape of a club would count as both an improvised weapon and a simple club. If it breaks, it might become an improvised dagger (also a Kensai weapon).